| Time | Variable |
|---|---|
| 1 | 2.1 |
| 2 | 4.5 |
| 3 | 5.3 |
| 4 | 3.2 |
| 5 | 1.6 |
| Time | Variable | Lag |
|---|---|---|
| 1 | 2.1 | |
| 2 | 4.5 | 2.1 |
| 3 | 5.3 | 4.5 |
| 4 | 3.2 | 5.3 |
| 5 | 1.6 | 3.2 |
| Time | Variable | Lag |
|---|---|---|
| 1 | 2.1 | |
| 2 | 4.5 | 2.1 |
| 3 | 5.3 | 4.5 |
| 4 | 3.2 | 5.3 |
| 5 | 1.6 | 3.2 |
Correlation of a variable with its lag is first-order autocorrelation.
| Time | Variable | 2nd Lag |
|---|---|---|
| 1 | 2.1 | |
| 2 | 4.5 | |
| 3 | 5.3 | 2.1 |
| 4 | 3.2 | 4.5 |
| 5 | 1.6 | 5.3 |
Correlation of a variable with its 2nd lag is second-order autocorrelation.
Rearrange \[x_t = \alpha + \beta x_{t-1} + \varepsilon_t\] as \[\Delta x_t = \alpha + (\beta-1) x_{t-1} + \varepsilon_t\] \[\Delta x_t = (\beta-1)\left(x_{t-1} - \frac{\alpha}{1-\beta}\right) + \varepsilon_t\]
An AR(1) is \[\Delta x_t = (\beta-1)\left(x_{t-1} - \frac{\alpha}{1-\beta}\right) + \varepsilon_t\]
\(\beta<1\) implies reregression towards the mean.
The mean is \(\alpha/(1-\beta)\).
\(\beta>1\) implies nonstationary.
Ask Julius to plot the ACF, fit an AR model, and forecast the following:
Percent change in crude price: Ask Julius to use pandas-datareader to download crude oil prices starting in 1980 from FRED, to downsample to end-of-month, and to compute percent changes.
Change in Treasury yields: Ask Julius to use pandas-datareader to download 10-year Treasury yields starting in 1980 from FRED, to downsample to monthly, and to compute changes.